Chern-Simons terms, �chiral anomalies and tachyons
Edwan PREAU
07/07/22
Collaborators: Elias KIRITSIS (APC), Francesco NITTI (APC), Matti Järvinen (APCTP)
To appear soon on the ArxiV
12th Crete Regional Meeting in String Theory
Motivation
Coupling to external fields
Without external fields : CS term contributes to 3-point functions and higher
External fields : 2-point functions
CS terms responsible for the chiral anomalies of the boundary theory in presence of external fields
Baryonic Physics
In holographic QCD models, the CS term is known to be necessary to construct a bulk solution dual to a baryon state
Dynamically : stabilizes the baryon size
[hep-th/9905111]
[0806.3122]
[0807.0316]
The Role of the CS action in holography
Holographic QCD
Semi-classical limit of a QG theory on a higher-D space-time (bulk)
Strongly coupled QCD-like 4D QFT
Phenomenological approach : bottom-up holography
🡪 Most complete framework for bottom-up holographic QCD : V-QCD
[1112.1261]
Outline
Holographic QCD : dictionary
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The main operators relevant to the vacuum structure of low-energy QCD are
Color
Flavor
Holographic QCD : field content
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The main operators relevant to the vacuum structure of low-energy QCD are
Color
Flavor
Holographic QCD : the WSS model
COLOR
| 0 | 1 | 2 | 3 | (4) | 5 | 6 | 7 | 8 | 9 |
| • | • | • | • | • | | | | | |
[hep-th/0412141]
Holographic QCD : the WSS model
FLAVOR
| 0 | 1 | 2 | 3 | (4) | 5 | 6 | 7 | 8 | 9 |
| • | • | • | • | • | | | | | |
D8 | • | • | • | • | | • | • | • | • | • |
SO(5) : project on singlets
[hep-th/0412141]
[hep-th/0412141]
Consequence : Single gauge field
External gauge fields in the boundary theory:
WSS CS term and QCD chiral anomaly
[hep-th/0412141]
QCD chiral anomaly
The bottom-up V-QCD framework
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Color
Flavor
The V-QCD Framework : Action
The V-QCD action is built by deforming what is known from top-down holography with phenomenological parameters of the bulk theory
Parameters of the bulk theory | |
| |
The V-QCD Framework : Action
The V-QCD action is built by deforming what is known from top-down holography with phenomenological parameters of the bulk theory
[hep-th/0303057]
[hep-th/0012210]
Sen :
The V-QCD Framework : Action
The V-QCD action is built by deforming what is known from top-down holography with phenomenological parameters of the bulk theory
Parameters of the bulk theory | ||
| | |
V-QCD :
[1112.1261]
Previous Results in V-QCD
Baryon Density
Temperature
V-QCD phase diagram
[1112.1261]
[1210.4516]
[1312.5199]
Top-down ST models include CP-odd topological Chern-Simons (CS) terms
The CS part of the bulk action is necessary for chiral anomalies and baryonic physics
🡪 What should be the form of the CS term in the bottom-up V-QCD action ?
The CS part of the V-QCD action
In ST, the tachyon dependence is known only in the maximally supersymmetric case.
[hep-th/0012210]
[hep-th/0702155]
V-QCD CS action : closed part
[Witten, 1983]
V-QCD CS action : non-closed part
[hep-th/0303057]
[hep-th/0012210]
V-QCD baryon
The bulk instanton
Quantization of isospin 0-modes : baryon spectrum in the V-QCD model
Properties of 1-baryon solution = input for the construction of a dense baryonic matter approximate solution 🡪 relevant for NS matter
Equations of motion for the fields of the ansatz are solved numerically
Summary
Appendix
Holographic QCD : two approaches
Top-Down holography
Construct an explicit QG dual (string theory)
that looks like QCD at low-energy
Typically involves brane systems in 10-D
Bottom-Up holography
Holographic dictionary : bulk field content dual to the main operators relevant to low-energy QCD
Construct an EFT for the bulk fields :
Phenomenological approach : bottom-up 🡪 Most complete framework : V-QCD
[1112.1261]
Arguments for the selection of operators in holographic QCD
In the UV, operators with higher scaling dimension and/or spin have higher mass dual fields in the bulk
This corresponds to the result that the RG flow equations near a CFT can be truncated to consider only the flows for the most relevant operators
In the IR, higher-dimension operators can become important in theories that are strongly coupled in the IR
There are indications that such operators are the dual of higher level excitations of the string
[hep-th/9905111]
B. Chiral Lagrangian, WZW term and QCD chiral anomalies
Chiral Effective Field Theory
28/32
One approach that has proven successful in the study of QCD at low energy is that of the chiral effective field theory
QCD possesses an approximate symmetry : the chiral symmetry
29/32
The WZW term
The gauged WZW term
C. The CS action from the WZ sector of the branes action
Brane tension
[hep-th/0702155] – supermatrix formalism
🡪 Coupling between the worldvolume flavor fields and the Type II RR forms
[hep-th/0012210]
D. Baryons in Holography
35/34
SYM
The internal space is important for the construction of the baryon in SYM
Holographic Baryons in SYM
36/34
N
-1
-1
-1
-1
Charge under the U(1) brane world-volume gauge field
CS action and Baryon Number in the SS model
37/32
E. Details about the V-QCD potentials
A comment on the DBI form of the action
39/46
[1112.1261]
Details about the potentials ansätze: glue
40/46
Parameters of the bulk theory | |
| |
[1809.07770]
Details about the potentials ansätze: glue
41/46
Fit to lattice YM thermodynamics
42/46
The solid lines and error bars represent the extrapolation of lattice data to Nc = ∞ and the dashed curves are the results for the holographic model.
[1809.07770]
43/46
[0907.3719]
Details about the potentials ansätze: DBI
44/46
Parameters of the bulk theory | ||
| | |
We parametrize :
[1112.1261]
45/46
46/46
47/46
Fit to lattice QCD thermodynamics
48/46
[1809.07770]
Closing remarks on the potentials
49/46
Once the UV and IR boundary conditions are fixed, 12 parameters are used to fit lattice data
The fit is rigid : dependence on all parameters is weak
The shape of the curves is a prediction of holography
The resulting potentials are actually simple monototic functions
[2110.08281]
F. The gauge-invariant CS forms
G. The V-QCD baryon
The Bulk Instanton
53/34
z
SO(3)
The Bulk Instanton : ansatz
54/34
SU(2)
U(1)
CS
The Bulk Instanton : Numerical Solution
55/34
z
Quantization of the collective modes
The baryon states are found by quantizing the collective modes of the classical instanton background
56/34
Collective Modes |
Translation in 3D |
|
Translation in z |
Dilation, … |
0-modes
The Rotating Soliton
57/34
Solution of the classical bulk EoMs corresponding to a rotating soliton
The rotation is a perturbation if
Soliton moment of inertia
Soliton mass
Static
Rotating
The Rotating Soliton : Quantization
58/34
The solution is obtained numerically